Question

let Xn be a sequence in a metric space X . If Xn -> x in X iff every neighbourhood of x contains all but finitely many points of the terms of {Xn}

Answer #1

is about metric spaces:
Let X be a metric discret space show that a sequence x_n in X
converge to l in X iff x_n is constant exept for a finite number of
points.

Let (X,d)
be a complete
metric space, and T
a d-contraction
on X,
i.e., T:
X
→ X
and there exists a q∈
(0,1) such that for all x,y
∈ X,
we have d(T(x),T(y))
≤ q∙d(x,y).
Let a
∈ X,
and define a sequence (xn)n∈Nin
X
by
x1 :=
a
and ∀n ∈
N: xn+1
:= T(xn).
Prove, for all n
∈ N,
that d(xn,xn+1)
≤ qn-1∙d(x1,x2).
(Use
the Principle of Mathematical Induction.)
Prove that (xn)n∈N
is a d-Cauchy
sequence in...

Let (X,d) be a metric space. Let E ⊆ X. Consider the set L of
all points in X which are limits of sequences contained in E. Prove
or disprove the following:
(a) L⊆E. (b) L⊆Ē. (c) L̄ ⊆ Ē.

Suppose (an), a sequence in a metric space X, converges to L ∈
X. Show, if σ : N → N is one-one, then the sequence (bn = aσ(n))n
also converges to L.

Let {xn} be a non-decreasing sequence and assume that xn goes to
x as n goes to infinity. Show that for all, n in N (naturals), xn
< x. Formulate and prove an analogous result for a
non-increasing sequences.

Let
( xn) and (yn) be sequence with xn converge to x and yn converge to
y. prove that for dn=((xn-x)^2+(yn-y)^2)^(1/2), dn converge to
0.

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Let X = (xn) be a sequence in R^p which is convergent
to x. Show that lim(||xn||) = ||x||. hint: use triange
inequality

Let (X, d) be a compact metric space and let A ⊆ X. Suppose that
A is not compact. Prove that there exists a continuous function f :
A → R, from (A, d) to (R, d|·|), which is not uniformly
continuous.

Let xn be a sequence such that for every m ∈ N, m ≥ 2 the
sequence limn→∞ xmn = L. Prove or provide a counterexample: limn→∞
xn = L.

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